Nearly everyone roots for an underdog, While this devise has some similarities to the rivalry kind of element in a plot. There is one major difference. The hero or heroine isn't equal to the villain of the piece. They are for some reason with little power. But we do root for them to win.
When attempting this think of your reader. Play to their emotions. They want the underdog to win and they have little or no love for the villain. Show this character in ways that make your reader angry but not angry enough for them to toss the book aside. Let little grains of hope creep into the underdog's character. This is the good guy and he will win but not before suffering defeat at the hands of the bad guy. Here the rule of three often comes into play. We all know about three strikes and you're out. But using this devise means two strikes and the underdog hits a home run.
When the hero or heroine wins the battle, let the reader feel the triumph and show the underdog triumphant and the villain receiving his due justice.
This kind of devise can be used in other ways as in the man versus nature or man versus society. Just remember the underdog needs to triumph.